Chapter 15 I Prime Chart for Measurement Data With Variable Subgroup Sizes (and more)
The I chart is often regarded as the Swiss Army knife of SPC. It is useful in many situations and can frequently serve as a valid – or even superior – alternative to other Shewhart SPC charts. This is partly because the I chart is based on the empirical variation in data, rather than on theoretical distributional assumptions, which may or may not be true.
However, the I chart does not account for variable subgroup sizes – that is, when the area of opportunity varies between samples, such as when the number of patients fluctuates from month to month. This produces straight control limits, which in some situations may generate false signals or fail to detect true ones. When the variation is small, this may not pose a problem, but often – not the least in healthcare – size matters.
Several solutions have been proposed. Here we present the normalised I chart or, as we prefer to name it, the I prime (I’) chart suggested by Taylor (2018).
In summary:
The I’ chart accounts for variable subgroup sizes by adjusting the within subgroup standard deviation with a factor that depends on the size of each subgroup producing wavy control limits when subgroup sizes vary.
The I’ chart is useful for measurement and count data with or without denominators.
With constant subgroup sizes, the I’ chart is an exact match for the original I chart.
For rates and proportions, the I’ chart produces control limits that match those of the U' and P' charts.
Figure 15.1 shows an I’ and a P’ chart side-by-side demonstrating their similar results.
qic(month, deaths, admissions,
data = admis,
chart = 'ip',
y.percent = T,
title = "I' chart of hospital mortality",
xlab = 'Month')
qic(month, deaths, admissions,
data = admis,
chart = 'pp',
title = "P' chart of hospital mortality",
xlab = 'Month')
Figure 15.1: Comparing I’ and P’ charts
15.0.1 Procedure for calculating centre line and control limits
We use the following symbols:
- \(n\) = numerators
- \(d\) = denominators
- \(o\) = number of data values (subgroups)
- \(i\) = ith data value
Values to plot:
\[ y = \frac{n}{d} \]
Centre line:
\[ CL = \frac{\sum{n}}{\sum{d}} \]
Standard deviation of ith data point:
\[ s_i = \sqrt{\frac{\pi}{2}}\frac{\vert{}y_i-y_{i-1}\vert{}}{\sqrt{\frac{1}{d_i}+\frac{1}{d_{i-1}}}} \]
Average standard deviation:
\[ \bar{s} = \frac{\sum{s}}{o} \]
Control limits:
\[ \text{control limits} = CL \pm 3 \frac{\bar{s}}{\sqrt{d_i}} \]
When subgroup sizes equals 1, the control limits simplify to: \(CL \pm 2.66\overline{MR}\) as in the original I chart.
As with the original I chart, qicharts2 screens the moving ranges of \(s_i\), removing ranges greater than the theoretical upper natural limit (= 3.2665) before calculating \(\bar{s}\).
15.1 I’ charts for measument data with variable subgroup sizes
Figure 15.1 demonstrates the use of an I’ chart with count data. However, the I’ chart was originally intended for use with measurement data where subgroup sizes vary. For example, due to privacy considerations, we may only have access to averaged patient data rather than individual records. Traditionally, such data would be plotted on an original I chart, but if the number of patients in each subgroup varies significantly, straight control limits may be suboptimal.
Figure 15.2 is an I chart of the monthly HbA1c averages from the Diabetes HbA1c dataset. Notice the data point above the upper control limit in April 2020 suggesting a special cause. However, when plotting aggregated measurement data, the I chart does not account for variations in subgroup size.
qic(month, avg_hba1c,
data = diabetes,
chart = 'i',
title = NULL,
ylab = 'mmol / mol',
xlab = 'Month')
Figure 15.2: I chart of average HbA1c without denominator
The I’ chart in Figure 15.3 takes the subgroup size (number of patients) into account and adjusts the control limits correspondingly.
qic(month, avg_hba1c * n, n,
data = diabetes,
chart = 'ip',
title = NULL,
ylab = 'mmol / mol',
xlab = 'Month')
Figure 15.3: I prime chart of average HbA1c with denominator
April 2020 was the first month of lockdown during Covid-19 in Denmark, and the number of patients (the denominator) seen during this month was significantly lower than usual, which allowed for larger than usual common cause variation in measurements and consequently wider control limits this month. So, when the subgroup size is taken into account, the apparent special cause in Figure 15.2 is actually within the limits of the expected common cause variation.
Also, notice that the centre lines are a bit different (60.6 vs 60.3), because the I prime chart uses the weighted rather than the unweighted mean of the measurements.
Note that in order to plot the averages we need to multiply the numerators (avg_hba1c, which are already averaged) by the denominators (n) to get the sum of individual measurements to use as the new numerator. Otherwise, the result would be a plot of averages of averages.
15.2 One chart to rule them all?
The I’ chart produces results similar to the original I chart for individual measurements, and to the P’ and U’ charts for proportion and count data.
If we accept Laney’s claim that prime charts are valid substitutes for their non-prime counterparts, it is tempting to conclude that the I’ chart offers a universal solution to most, if not all, of our charting needs. Taylor largely supports this view – although he suggests that the X-bar chart is better suited for measurement data with subgroup sizes greater than one. We, however, see no reason that I’ charts cannot be used as an alternative to X-bar charts for data with multiple measurements per subgroup.
In conclusion, while the I’ chart is a recent development, its versatility and ability to mimic the behaviour of traditional SPC charts (including prime charts) make it a compelling alternative, that has the potential to streamline chart selection by serving as a single, robust option across various data types and scenarios. As the methodology continues to gain empirical support, the I’ chart may well become the preferred tool for modern quality control.